Bhatnagar-Gross-Krook Research Articles

Accelerated discrete velocity method for axial-symmetric gaseous flows

An accelerated discrete velocity method is presented for the simulation of axial-symmetric single-component gas flows in the whole range of gas rarefaction. The method relies on the Bhatnagar–Gross–Krook (BGK) kinetic equation. A diffusion-type moment equation is derived on the basis of the cylindrical kinetic one. Both equations are solved by discretizing the spatial and molecular velocity spaces in a joint iteration. The relationship between the cylindrical and the two-dimensional moment equations is shown. A stability analysis is developed to show the improved iteration performance of the accelerated method. Pressure-driven flow in a tube is simulated to study the method. The accelerated scheme is superior compared to the non-accelerated standard one at intermediate and high values of the rarefaction parameter in terms of the number of iterations required and the computational time. The flow rates and the velocity profiles are compared to the analytical discrete ordinate solution of the problems. There is a very good agreement between the results provided by the two methods.

Coupling Kinetic and Continuum Equations for Micro Scale Flow Computations

A hybrid method, coupling the direct numerical solution of the Bhatnagar-Gross-Krook (BGK) kinetic equation and hydrodynamic (Navier-Stokes) equations is presented. The computational physical domain is decomposed into kinetic and continuum sub-domains using an appropriate criterion based on the local Knudsen number and proper gradients of macro-parameters, computed via a preliminary Navier-Stokes solution throughout the whole physical domain. The coupling is achieved by matching half fluxes at the interface of the kinetic and Navier-Stokes domains, thus taking care of the conservation of momentum, energy, and mass through the interface. The advantage of the presented hybrid algorithm is that it easily allows the coupling of existing codes for the numerical solution of the BGK and Navier-Stokes equations. To validate and estimate the efficiency of the proposed method the simulation of the monatomic gas flow through a slit has been considered for outlet to inlet pressure ratio of 0.1, 0.5, and 0.9, and a wide range of Knudsen number. The comparison of local parameters (density, velocity, and temperature) with pure kinetic solutions shows satisfactory agreement with those computed by the hybrid solver.

On the Stability of the Finite Difference based Lattice Boltzmann Method

This paper is devoted to determining the stability conditions for the finite difference based lattice Boltzmann method (FDLBM). In the current scheme, the 9-bit two-dimensional (D2Q9) model is used and the collision term of the Bhatnagar- Gross-Krook (BGK) is treated implicitly. The implicitness of the numerical scheme is removed by introducing a new distribution function different from that being used. Therefore, a new explicit finite-difference lattice Boltzmann method is obtained. Stability analysis of the resulted explicit scheme is done using Fourier expansion. Then, stability conditions in terms of time and spatial steps, relaxation time and explicitly-implicitly parameter are determined by calculating the eigenvalues of the given difference system. The determined conditions give the ranges of the parameters that have stable solutions.

A new gas-kinetic scheme based on analytical solutions of the BGK equation

Up to an arbitrary order of the Chapman–Enskog expansion of the kinetic Bhatnagar–Gross–Krook (BGK) equation, a corresponding analytic solution can be obtained. Based on such a compact exact solution, a new gas-kinetic scheme is constructed for the compressible Navier–Stokes equations. Instead of using a discontinuous initial condition in the gas-kinetic BGK–NS method Xu [K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations, and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (2001) 289–335.], the new scheme starts with a continuous initial flow distribution at a cell interface which is obtained through an upwind-biased WENO reconstruction, and uses the time accurate solution for the flux evaluation. The new kinetic scheme not only preserves favorable properties of the existing BGK–NS method, such as stability, high resolution, and good performance in capturing discontinuity, but also achieves a very high efficiency, which is even more efficient than the same order well-defined classical finite difference scheme based on the macroscopic governing equations directly. The stability, accuracy, and efficiency of the new scheme are evaluated quantitatively through numerical tests. The new scheme captures sharp discontinuity without post-shock oscillation, and has high accuracy in resolving viscous solution. Due to the use of time-accurate analytical solution, the overall performance of the new scheme is superior in comparison with the finite difference or finite volume schemes which start from the same initial reconstruction and use the numerical Runge–Kutta methods for time accuracy.

A gas‐kinetic scheme for the modified Baer–Nunziato model of compressible two‐phase flow

SUMMARYNumerical methods for the Baer–Nunziato model of compressible two‐phase flow have attracted much attention in recent years. In this paper, a two‐phase Bhatnagar–Gross–Krook (BGK) model is constructed in which the non‐conservative terms in the Baer–Nunziato model are considered as the external forces and the collisions both with particles of their phases and other phases are taken into consideration. On the basis of this BGK model, the so‐called modified Baer–Nunziato model is derived and a gas‐kinetic scheme for this modified model is presented. The distribution functions are constructed at the cell interface based on the integral solutions of the BGK equations for both phases. Then, numerical fluxes can be obtained by taking moments of the distribution functions, and non‐conservative terms are explicitly introduced into the construction of numerical fluxes. In this method, not only the iterative processes in the exact Riemann solvers are eliminated but also the collisions with the particles of other phases are taken into account. Copyright © 2012 John Wiley & Sons, Ltd.

Diffusive Limits for a Quantum Transport Model with a Strong Field

We derive semiclassical diffusive equations for the local electron densities in a semiconductor characterized by a two-band k·p Hamiltonian, under the action of a strong external field. By using a spinorial formalism, we consider the quantum kinetic (Wigner) system endowed with a Bhatnagar-Gross-Krook (BGK)-like interaction term. Diffusive equations are derived by the Chapman-Enskog method. The closure of such equations is obtained by using the quantum version of the minimum entropy principle. In practice, it is unfeasible to put in an explicit form the diffusive equations in the general case, even in the semiclassical limit. Then we investigate the case in which band parameters have little influence on the dynamics at the macroscopic scale.

Simulation of Mono- and Bidisperse Gas-Particle Flow in a Riser with a Third-Order Quadrature-Based Moment Method

Gas-particle flows can be described by a kinetic equation for the particle phase coupled with the Navier–Stokes equations for the fluid phase through a momentum exchange term. The direct solution of the kinetic equation is prohibitive for most applications due to the high dimensionality of the space of independent variables. A viable alternative is represented by moment methods, where moments of the velocity distribution function are transported in space and time. In this work, a fully coupled third-order, quadrature-based moment method is applied to the simulation of mono- and bidisperse gas-particle flows in the riser of a circulating fluidized bed. Gaussian quadrature formulas are used to model the unclosed terms in the moment transport equations. A Bhatnagar–Gross–Krook (BGK) collision model is used in the monodisperse case, while the full Boltzmann integral is adopted in the bidisperse case. The predicted values of mean local phase velocities, rms velocities, and particle volume fractions are compared with the Euler–Lagrange simulations and experimental data from the literature. The local values of the time-average Stokes, Mach, and Knudsen numbers predicted by the simulation are reported and analyzed to justify the adoption of high-order moment methods as opposed to models based on hydrodynamic closures.

Analysis of Coupled Lattice Boltzmann Model with Well-Balanced Scheme for Shallow Water Flow

The coupled lattice Boltzmann method (CLBM) is applied in investigating contamination transport in shallow water flows. Shallow water equations and advection-diffusion equation are both solved using lattice Boltzmann method (LBM) on a D2Q9 square lattice and Bhatnagar-Gross-Krook (BGK) term. For extending application of CLBM in shallow water flows, the well-balanced scheme is introduced to replace the source term. Three cases including dam break, 2D pure diffusion and complex tidal flow are calculated and analyzed. Dam break and 2D pure diffusion are prepared to validate the flow module and water quality module, respectively. Both the cases show satisfactory consistency between predicting results and analytical solutions. Since clear reproduction of the shock wave propagation and precise prediction of contamination transport are derived, LBM is proved to be the numerical method naturally conservative with acceptable computing error. Furthermore, complex tidal flow with irregular geometry and sinus-varied bathymetry is simulated by adopting the well-balanced treating on the source term. The velocity fields, water levels, and water quality are compared between the ebb tide and flood tide, the results of which are in excellent accordance with the physical laws during the process. Hence, it may demonstrate that improved by well-balanced scheme CLBM can be widely applicable in shallow water flow.

A gas kinetic scheme for the Baer–Nunziato two-phase flow model

Numerical methods for the Baer–Nunziato (BN) two-phase flow model have attracted much attention in recent years. In this paper, we present a new gas kinetic scheme for the BN two-phase flow model containing non-conservative terms in the framework of finite volume method. In the view of microscopic aspect, a generalized Bhatnagar–Gross–Krook (BGK) model which matches with the BN model is constructed. Based on the integral solution of the generalized BGK model, we construct the distribution functions at the cell interface. Then numerical fluxes can be obtained by taking moments of the distribution functions, and non-conservative terms are explicitly introduced into the construction of numerical fluxes. In this method, not only the complex iterative process of exact solutions is avoided, but also the non-conservative terms included in the equation can be handled well.

Numerical simulation of micro flows with moving obstacles

We present a numerical method for computing unsteady rarefied gas micro flows, in domains with moving boundaries, in view of applications to complex computations of moving structures in micro or vacuum systems. The flow is described by the Bhatnagar-Gross-Krook (BGK) model of the Boltzman equation. A standard approach to simulate incompressible viscous flows with moving boundaries is the immersed boundary method. In this work, we propose an extension of this approach to a deterministic simulation of rarefied flows. The immersed boundary approach consists in keeping the same mesh all along the calculation: every cell of the mesh remains fixed for all time steps while the domain occupied by the gas changes. This strategy avoids to use moving meshes and remeshing approaches, and should be easily applied to problems with complex geometries. The method has been tested with both specular and diffuse boundary conditions and has been validated on several 1D problems (moving piston, actuator, etc.). Up to our knowledge, this is the first time that a deterministic simulation method for rarefied flows with moving obstacles based on the immersed boundary method is presented.

Flow of power-law fluids in a cavity driven by the motion of two facing lids – A simulation by lattice Boltzmann method

The study of a non-Newtonian fluid flow behavior in mixing cavities is of great importance in process industries. In the present work, a Bhatnagar–Gross–Krook (BGK) approximation based lattice Boltzmann method (LBM) is presented to simulate non-Newtonian power law fluid flows in a double sided lid driven cavity. First, the code is validated against numerical results taken from the published sources for power law fluid flow in cavity driven by the uniform motion of lid. Next, the code is applied for two different cases-parallel wall motion and anti-parallel wall motion of two sided lid driven cavity. The influence of power law index (n) and Reynolds number (RePL) on the variation of velocity and center of vortex location of fluid has been analyzed with the help of velocity profiles and streamline plots. Additionally, we also study the effect of speed ratio on development of vortex in a cavity. Further, the effect of n on variation of drag coefficient has been presented. Finally, we present the location of vortex center and computational time required for a system to reach the steady state.

High accuracy numerical solutions of the Boltzmann Bhatnagar-Gross-Krook equation for steady and oscillatory Couette flows

Modeling gas flows generated by micro- and nano-devices often requires the use of kinetic theory. To facilitate implementation, various approximate formulations have been proposed based on the Bhatnagar-Gross-Krook (BGK) kinetic model, including most recently, the lattice Boltzmann (LB) method. While there exists a comprehensive numerical data set for the hard sphere linearized Boltzmann equation for steady Couette flow, no such set of data is available for the Boltzmann-BGK equation. The purpose of this article is to present a high accuracy data set for the linearized Boltzmann-BGK equation over the full range of Knudsen numbers and normalized oscillation frequencies – this encompasses both steady and unsteady Couette flows. This data set is expected to be of particular value in the benchmarking and validation of computational methods such as the LB method and other approaches based on the Boltzmann-BGK equation.

Pressure driven rarefied gas flow through a slit and an orifice

The flow of a monatomic gas through a slit and an orifice due to an arbitrarily large pressure difference is examined on the basis of the nonlinear Bhatnagar–Gross–Krook (BGK) model equation, subject to Maxwell diffuse boundary conditions. The governing kinetic equation is discretized by a second-order control volume scheme in the physical space and the discrete velocity method in the molecular velocity space. The nonlinear fully deterministic algorithm is optimized to reduce the computational effort by introducing memory usage optimization, grid refinement and parallelization in the molecular velocity space. Results for the flow rates and the macroscopic distributions of the flow field are presented in a wide range of the Knudsen number for several pressure ratios. The effect of the various geometric and physical parameters on the flow field is examined. Comparison with previously reported corresponding Direct Simulation Monte Carlo (DSMC) results indicates a very good agreement, which clearly demonstrates the accuracy of the kinetic algorithm and furthermore the reliability of the BGK model for simulating pressure driven flows.

An implicit lattice Boltzmann model for heat conduction with phase change

In this work, a new variation of the lattice Boltzmann method (LBM) was developed to solve the heat conduction problem with phase change. In contrast to previous explicit algorithms, the latent heat source term was treated implicitly in the energy equation, avoiding iteration steps and improving the formulation stability and efficiency. The Bhatnagar–Gross–Krook (BGK) approximation with a D2Q9 lattice was applied and different boundary conditions including Dirichlet and Neumann boundary conditions were considered. The developed model was tested by solving a one-dimensional melting problem for a pure metal, and one and two-dimensional solidification problems for a binary alloy. The results of the LBM solution were compared with analytical and finite element solutions and a good consistency was observed. Considering the special capabilities that LBM offers, like local characteristic, and inherent parallel structure, the developed model is an interesting alternative to traditional continuum models.

Rarefied gas flow around a sharp edge induced by a temperature field

AbstractA rarefied gas flow thermally induced around a heated (or cooled) flat plate, contained in a vessel, is considered in two different situations: (i) both sides of the plate are simultaneously and uniformly heated (or cooled); and (ii) only one side of the plate is uniformly heated. The former is known as the thermal edge flow and the latter, typically observed in the Crookes radiometer, may be called the radiometric flow. The steady behaviour of the gas induced in the container is investigated on the basis of the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation and the diffuse reflection boundary condition by means of an accurate finite-difference method. The flow features are clarified for a wide range of the Knudsen number, with a particular emphasis placed on the structural similarity between the two flows. The limiting behaviour of the flow as the Knudsen number tends to zero (and thus the system approaches the continuum limit) is investigated for both flows. The detailed structure of the normal stress on the plate as well as the cause of the radiometric force (the force acting on the plate from the hotter to the colder side) is also clarified for the present infinitely thin plate.

Progress in gas-kinetic upwind schemes for the solution of Euler/Navier–Stokes equations – I: Overview

Three gas-kinetic upwind schemes for the solution of the Euler/Navier–Stokes equations are reviewed. They are Kinetic Flux-Vector Splitting (KFVS), Kinetic Wave/Particle Splitting (KWPS), and Bhatnagar–Gross–Krook (BGK) methods. For the Euler equations, the most sophisticated BGK scheme can be interpreted as a relaxation scheme between the two KFVS schemes with different moments, KFVS and KFVS_ u 0. It improves the accuracy over the KFVS scheme and the robustness over the KFVS_ u 0 scheme. The direct generalization of this relaxation approach to the Navier–Stokes equations leads to a much simpler BGK scheme than the one in the literature. In this simplified BGK scheme, there exist two types of particle collision time. The one in the BGK model acts as a relaxation parameter. Its role is to add some numerical dissipation from the KFVS scheme to the KFVS_ u 0 scheme. On the other hand, the one in the Chapman–Enskog expansion of the gas distribution function is related to physical dissipation. Following the same approach, another type of BGK schemes is further developed, which is a relaxation scheme between KWPS and KWPS_ u 0. In spite of the fact that the KWPS scheme is more diffusive than the KFVS scheme, a BGK scheme based on KWPS and KWPS_ u 0 is found not only computationally more efficient but also less diffusive than a BGK scheme based on KFVS and KFVS_ u 0. However, this issue needs further and more rigorous investigation by performing the numerical analysis of a model 1-D convection–diffusion equation.

Comment on “Heat transfer and fluid flow in microchannels and nanochannels at high Knudsen number using thermal lattice-Boltzmann method”

In this Comment we reveal the falsehood of the claim that the lattice Bhatnagar-Gross-Krook (BGK) model "is capable of modeling shear-driven, pressure-driven, and mixed shear-pressure-driven rarified [sic] flows and heat transfer up to Kn=1 in the transitional regime" made in a recent paper [Ghazanfarian and Abbassi, Phys. Rev. E 82, 026307 (2010)]. In particular, we demonstrate that the so-called "Knudsen effects" described are merely numerical artifacts of the lattice BGK model and they are unphysical. Specifically, we show that the erroneous results for the pressure-driven flow in a microchannel imply the false and unphysical condition that 6σKn<-1, where Kn is the Knudsen number σ=(2-σ(v))/σ(v) and σ(v)∈(0,1] is the tangential momentum accommodation coefficient. We also show explicitly that the defects of the lattice BGK model can be completely removed by using the multiple-relaxation-time collision model.

An upwinded state approximate Riemann solver

Stability is achieved in most approximate Riemann solvers through ‘flux upwinding’, where the flux at the interface is arrived at by adding a dissipative term to the average of the left and right flux. Motivated by the existence of a collapsed interface state in the gas-kinetic Bhatnagar–Gross–Krook (BGK) method, an alternative approach to upwinding is attempted here; an interface state is arrived at by taking an upwinded average of left and right states, and then the flux is calculated as a function of this ‘collapsed’ interface state. This so called ‘state-upwinding’ approach gives rise to a new scheme called the linearized Riemann solver for the Euler and Navier–Stokes equations. The scheme is shown to be closely associated with the Roe scheme. It is, however, computationally less expensive and gives qualitatively comparable results over a wide range of problems. Most importantly, this scheme is found to preserve stationary contacts while not exhibiting the carbuncle phenomenon which plagues the Roe and other contact-preserving schemes. The scheme is therefore motivated as a new starting point to analyze the origin of the carbuncle phenomenon. Copyright © 2011 John Wiley & Sons, Ltd.

Fluctuations in collisional plasma in the presence of an external electric field

The theory of large-scale fluctuations in a plasma is used to calculate the correlations functions of electron and ion density with regard to particle collisions described within the Bhatnagar-Gross-Krook (BGK) model and the presence of a constant external electric field. The changes of plasma particle distribution functions due to an external electric field and their influence on the plasma dielectric response are taken into account. The dispersion relations for longitudinal waves in such a plasma are studied in details. It is shown that external electric field can lead to the ion-acoustic wave instability and anomalous growth of the fluctuation level. Detailed numerical studies of the general relations for electron number density fluctuations are performed and the effect of external electric field on the fluctuation spectra is studied.

Flux Limiter Lattice Boltzmann Scheme Approach to Compressible Flows with Flexible Specific-Heat Ratio and Prandtl Number

We further develop the lattice Boltzmann (LB) model [Physica A 382 (2007) 502] for compressible flows from two aspects. Firstly, we modify the Bhatnagar—Gross—Krook (BGK) collision term in the LB equation, which makes the model suitable for simulating flows with different Prandtl numbers. Secondly, the flux limiter finite difference (FLFD) scheme is employed to calculate the convection term of the LB equation, which makes the unphysical oscillations at discontinuities be effectively suppressed and the numerical dissipations be significantly diminished. The proposed model is validated by recovering results of some well-known benchmarks, including (i) The thermal Couette flow; (ii) One- and two-dimensional Riemann problems. Good agreements are obtained between LB results and the exact ones or previously reported solutions. The flexibility, together with the high accuracy of the new model, endows the proposed model considerable potential for tracking some long-standing problems and for investigating nonlinear nonequilibrium complex systems.